SECT. VI.] GENEKAL EQUATIONS OF PROPAGATION. 113 



that is to say, to add to this expression its own differential taken 

 with respect to z only ; we then have 



Kdx dtj -y- dt Kdx d u ^ dz 

 J dz * dz 



as the value of the quantity which escapes across the upper 

 rectangle. The same molecule receives also across the first 

 rectangle dz dx which passes through the point m, a quantity 



of heat equal to K-j- dz dx dt ; and if we add to this ex 

 pression its ow r n differential taken with respect to y only, we 

 find that the quantity which escapes across the opposite face 

 dz dx is expressed by 



K-j- dz dx dt K . ^ dy dz dx dt. 

 y y 



Lastly, the molecule receives through the first rectangle dy dz 

 a quantity of heat equal to K -y- dy dz dt, and that which it 



CiX 



loses across the opposite rectangle which passes through m is 

 expressed by 



,^ 777 -rr dX 7777 



K-r dy dzdtK -r dx dy dz dt. 



We must now take the sum of the quantities of heat which 

 the molecule receives and subtract from it the sum of those 

 which it loses. Hence it appears that during the instant dt, 

 a total quantity of heat equal to 



accumulates in the interior of the molecule. It remains only 

 to obtain the increase of temperature which must result from 

 this addition of heat. 



D being the density of the solid, or the weight of unit of 



volume, and C the specific capacity, or the quantity of heat 



which raises the unit of weight from the temperature to the 



temperature 1 ; the product CDdxdydz expresses the quantity 



F. H. 8 



